Remark

In fact, an enrichment over pointed sets consists precisely of the choice of a ‘zero’ morphism 0c,d:c→d0_{c,d}:c\to d for each pair of objects, with the property that 0c,d∘f=0b,d0_{c,d} \circ f = 0_{b,d} and f∘0a,b=0a,cf\circ 0_{a,b} = 0_{a,c} for any morphism f:b→cf:b\to c. Such an enrichment is unique if it exists, for if we are given a different collection of zero morphisms 0′c,d0'_{c,d}, we must have

0′c,d=0′c,d∘0c,c=0c,d0'_{c,d} = 0'_{c,d} \circ 0_{c,c} = 0_{c,d}

for any c,dc,d. Thus, the existence of zero morphisms can be regarded as a property of a category, rather than structure on it. (To be more precise, it is an instance of property-like structure, since not every functor between categories with zero morphisms will necessarily preserve the zero morphisms, although an equivalence of categories will.)